Symmetric rank 1 update

A. R. Conn, N. I. M. Gould, and Ph. L. Toint, Math. Prog., 2, 177 (1991). Convergence of Quasi-Newton Matrices Generated by the Symmetric Rank One Update. 

One of the most successful and widely used updating formulas is known as BFGS for its four developers Broyden, Fletcher, Goldfarb, and Shanno.6 95 It is a rank 2 update with inherent positive-definiteness (i.e., Bk positive-definite Bk+j positive-definite) that was derived bj symmetrizing the Broyden rank 1 update.5 6 95 A sequence of matrices B is generated from a positive-definite B0 (which may be taken as the identity) by the BFGS formula... 

Our companion book (Buzzi-Ferraris and Manenti, 2010a) discusses the updating procedure of L and D that factorizes a symmetric positive definite matrix to which a rank-1 ss -type matrix is added. Obviously, since two of such matrices are added sequentially here, this procedure should be repeated twice. It is important to know that the new factorization requires 0(wy) calculations. 

The MS-update (see Eq. (17)) is not a least-change update, i.e. it cannot be obtained as a solution of the constrained minimization problem (14), but it belongs to another important class of updates, namely to the Broyden s class (see below). A symmetric least-change update must be at least of rank two. 

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